Lines, Distances & Conic Sections

Analytic geometry questions are short if you recognise the form. Most are one of three things: a line through two points, the distance between them, or the centre and radius of a circle hidden in an expanded equation.

Worked: a circle in disguise

x² + y² − 6x + 4y − 12 = 0. Complete the square in each variable: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4, so (x − 3)² + (y + 2)² = 25. Centre (3, −2), radius 5.